PSY201: Chapter 7 - Sample Means Flashcards
Samples and Populations
can choose 1000s of potential samples from pop - have to worry about sampling error
diff samples will differ from each other ⇒ won’t necessarily have same sample means
Distribution of Sample Means
set of possible samples forms pattern⇒ distribution of sample means - allows us to predict sample characteristics
set of sample means for all the possible random samples of specific size (n) that can be selected from a pop
Distribution of Sample Means
we need to make inferences about sample mean rather than single score
distribution of sample means is diff from distribution of scores because sample means are statistics ⇒ sampling distribution
Distribution of Sample Means
statistic obtained by selecting all possible samples of specific size from a pop
sample means pile up around pop mean because they are representative of pop
Distribution of Sample Means
pile of sample means tends to be normally distributed
rare to find sample means really far from μ
larger the sample size⇒ closer sample means typically are to μ
Distribution of Sample Means
very important distribution
spread of sampling distribution of mean decreases as sample size increases
Distribution of Sample Means
mean of the distribution is not affected by sample size
The Central Limit Theorem
distribution of sample means has well-defined, predictable characteristics specified in the Central Limit Theorem:
Given distribution with a mean μ + standard deviation σ
The Central Limit Theorem
sampling distribution of the mean approaches normal distribution with a mean of μ + standard deviation of σ/√n, as n approaches infinity n = sample size # of samples assumed to be infinite.
The Central Limit Theorem
Sample mean distributions approach a normal distribution very quickly as n increases, even if original distribution is not normal
The Central Limit Theorem
assume sample mean distribution is normal if either:
pop from which samples are obtained is normal
sample size is n>/=30
The Central Limit Theorem
expect sample means to be close to pop mean
sample means should “pile up” around μ
distribution of sample means tends to form a normal shape
The Central Limit Theorem
mean of distribution of sample means = Expected Value of M
always = μ
The Central Limit Theorem
individual sample mean will probably not be identical to its pop mean; some error between M and μ.
Some sample means relatively close to μ, others will be relatively far away
The Central Limit Theorem
standard distance between M and μ = Standard Error of M
standard deviation of distribution
The Central Limit Theorem
standard error of M measures how well sample mean represents entire distribution
σ2M = σ2/n
SEM=σM = σ/√n
Standard Error
describes distribution of sample means
how much difference is expected from 1 sample to another
Standard Error
measures how well individual sample mean represents entire distribution
how much distance is reasonable to expect betw a sample mean + pop mean for distribution of sample means
Standard Error
population standard deviation, σ measures standard distance betw a single score (X) + population mean.
standard deviation provides a measure of the error expected for smallest possible sample, when n = 1
Standard Error
As the sample size increases, ⇒ error should decrease
larger the sample, more accurately should represent its population
law of large numbers
larger the sample size, n, more probable it is sample mean will be close to pop mean.
Standard Error
Determined by sample size
Determined by standard deviation of pop
if you have a lot of variability in pop ⇒ higher SEM
z-score
can be computed for each sample mean
tells where specific sample mean located relative to all other sample means
must remember to consider the sample size (n) + compute SE before you start any other computations
Probability and the Distribution of Sample Means
distribution of sample means must satisfy at least 1 of criteria for having normal shape before you can use unit normal table
Probability and the Distribution of Sample Means
normal sample mean distribution ⇒ can describe how particular sample related to another potential sample
an look up probabilities in unit normal table regarding likelihood of particular sample given makeup of pop
Probability and the Distribution of Sample Means
probability associated with specific sample mean = proportion of all possible sample means
Probability and the Distribution of Sample Means
z = M−μ/σM
Probability and the Distribution of Sample Means
positive z-score = M > μ + negative z-score = M < μ
numerical value of the z- score indicates distance betw M + μ in terms of standard error
Probability and the Distribution of Sample Means
procedure for finding probabilities for sample means is same as we used for individual scores
Looking Ahead to Inferential Statistics
there is always margin of error that we must consider when we use sample mean to make inferences about pop mean.
Looking Ahead to Inferential Statistics
Determine whether manipulated independent variable had effect on our dependent variable - sample mean for treatment group really different from sample mean for the control group
use distribution of sample means + standard error to help us make this decision
Looking Ahead to Inferential Statistics
use the distribution of sample means to show what we would expect scores to be for a control group
If treatment group is not noticeably different from expected mean distribution scores⇒ treatment did not have an effect
Looking Ahead to Inferential Statistics
use criteria of there being z = +/- 1.96
Looking Ahead to Inferential Statistics
standard error tells us how close diff sample means are clustered around true pop mean
small SE ⇒ most samples similar, doesn’t really matter which sample we get
Looking Ahead to Inferential Statistics
SE large ⇒ lot of deviation from sample to sample, sample might not be representative
we will want to replicate our results using diff samples to become more confident in findings
simple solution: obtain as large a sample as we can
remember: SE gets smaller when we increase n
